Initial-boundary value problem for an integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on a finite interval

In this paper, we explore the initial-boundary value (IBV) problem for an integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on the finite interval by extending the Fokas unified transform approach. The solution of this system can be expressed in terms of the solution of a 4x4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. Furthermore, the relevant jump matrices with explicit (x, t)-dependence of the matrix RH problem can be explicitly found via three spectral functions {s(k), S(k), S_L(k)} arising from the initial data and the Dirichlet-Neumann boundary conditions at x=0 and x=L, respectively. The global relation is also found to deduce two distinct but equivalent types of representations (i.e., one via the large $k$ of asymptotics of the eigenfunctions and another one in terms of the Gel'fand-Levitan-Marchenko (GLM) approach) for the Dirichlet and Neumann boundary value problems. In particular, the formulae for IBV problems on the finite interval can reduce to ones on a half-line as the length $L$ of the interval approaches to infinity. Moreover, we also present the linearizable boundary conditions for the GLM representations.